# Calculate/estimate the value of trigonometric functions in the unit circle by the value of their nearest angles

Given the fact that the values of trigonometric function (especially $sin$, $cos$, $tan$) for standard angles is easy to remember, is there a way that you can find / estimate the value of an angle in between?

For example given the values of $\frac{\pi}{2} (90^{\circ})$ and $\frac{4\pi}{6} (120^{\circ})$ is it possible to find / estimate the value of an angle in between? Let's say $\frac{5}{9\pi} (100^{\circ})$?

If you are close to a value (say $a$) for which you know the vaue of the trigonometric functions, you can use a Taylor expansion since $$\sin x=\sin (a)+(x-a) \cos (a)-\frac{1}{2} (x-a)^2 \sin (a)-\frac{1}{6} (x-a)^3 \cos (a)+O\left((x-a)^4\right)$$ For example $x=\frac{5\pi}{9}$ and we select $a=\frac{\pi}{2}$. So applying the above formula,$$\sin \frac{5\pi}{9}=1 -\frac{1}{2}\left(\frac{5 \pi }{9}-\frac{\pi}{2}\right)^2=1-\frac{\pi ^2}{648} =0.984769$$ while the exact value is $0.984808$.